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Theorems in Projective Geometry

  • Jeremy Gray
Part of the Springer Undergraduate Mathematics Series book series (SUMS)

Abstract

This chapter is more mathematical. The theorems of Pappus, Desargues, and Pascal are introduced to show that there is a non-metrical geometry such as Poncelet had described. However, they are proved in a modern way, by reducing them to simple special cases and then using a mixture of elementary vector methods and theorems from elementary Euclidean geometry. The reductions use simple projective transformations, which leads to discussion of the idea of augmenting the familiar plane with a line at infinity. The concept of the cross-ratio of four points on a line is introduced and shown to be invariant under a projective transformation. Poncelet’s porism is described.

Keywords

Projective Geometry Projective Transformation Conic Section Outer Conic Closure Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Department of MathematicsThe Open UniversityWalton Hall, Milton KeynesUnited Kingdom
  2. 2.The Mathematics InstituteThe University of WarwickWarwickUnited Kingdom

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